Trigonometry and its Applications
Trigonometry and its Applications covers ratios, identities and height–distance problems. The exam expects exact values, identity manipulation, and correct interpretation of angles of elevation and depression.
Core rules
- In a right triangle: sin=opposite/hypotenuse, cos=adjacent/hypotenuse, tan=opposite/adjacent; reciprocals cosec, sec, cot.
- Standard values: sin30=1/2, cos30=√3/2, tan30=1/√3; sin45=cos45=1/√2; sin60=√3/2, cos60=1/2, tan60=√3.
- Identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.
- Complementary angles: sin(90°−θ)=cosθ, tan(90°−θ)=cotθ.
- Angle of elevation is measured upward from horizontal; depression is measured downward from horizontal.
Worked example
A tower casts a shadow equal to its height. If the sun’s elevation is θ, then tanθ=height/shadow=1, so θ=45°. If the shadow were √3 times the height, tanθ=1/√3 and θ=30°.
Traps
Do not use the hypotenuse as the adjacent side. In depression problems, the angle at the object equals the angle at the observer only when the two horizontal lines are parallel. Keep degrees and radians separate. When simplifying identities, work on one side rather than crossing terms across the equals sign.